Here, \(P\)  is the projection onto the codespace \(\mathcal {C}\)  and \(\alpha _{ij}\)  are complex elements of a Hermitian matrix \(\alpha \) . The algebraic conditions in Eq. (REF ) are necessary and sufficient for the existence of a recovery (CPTP) map \(\mathcal {R}\)  which can correct perfectly for the errors \(\lbrace E_i\rbrace \) . We note here that information-theoretic conditions for perfect QEC have also been formulated in terms of the coherent information [1]}. In contrast to the active approach to QEC involving a recovery operation, there exist passive QEC strategies [2]}, where the information to be protected is stored in subspaces or subsytems of a larger Hilbert space that remain unaffected by the noise, called Decoherence-free subspaces (DFS) and Noiseless subsystems (NS) [3]}, respectively. Such passive QEC schemes are outside the scope of this review, here we will focus exclusively on active quantum error correction.
